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Mathematics 116 CHAPTER 1 Graphs of Equations Sketch the Graph of equations by Point Plotting

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Mathematics 116 CHAPTER 1 Graphs of Equations Sketch the Graph of equations by Point Plotting

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    1. Mathematics 116 CHAPTER 1 Graphs of Equations Sketch the Graph of equations by Point Plotting

    2. Objective: Graph an equation by using a graphing calculator.

    3. Graph an Equation with graphing calculator

    4. Graph an equation Circle with calculator

    5. T.S. Eliot - writer Where is the knowledge we have lost in information? Where is the wisdom we have lost in knowledge?

    6. Objective: Know the definition of a relation. A relation is a set of ordered pairs.

    7. Mathematics 116 Functions Note: Essential for the entire course!!!!!

    8. Know the definition of domain

    9. Know the definition of range.

    10. Graphs of Functions Vertical Line Test A set of points in a coordinate plane is the graph of y as a function of x if and only if no vertical line intersects the graph at more than one point.

    11. Mathematics 116 - 1.2 Lines in the Plane Objective: Find the slopes of lines

    12. Def: Linear Equation A linear equation in two variables is an equation that can be written in standard form ax + by = c where a,b,c are real numbers and a and b are not both zero.

    13. Def: Solution of linear equation in two variables A solution of a linear equation in two variables is a pair of numbers (x,y) that satisfies the equation. Ex:{(3,4)}

    14. Def: Intercepts y-intercept a point where a graph intersects the y-axis. x-intercept is a point where a graph intersects the x-axis.

    15. Procedure to find intercepts To find x-intercept 1. Replace y with 0 in the given equation. 2. Solve for x To find y-intercept 1. Replace x with 0 in the given equation. 2. Solve for y

    16. Find solutions to Equations with 2 variables 1. Choose a value for one of the variables 2. Replace the corresponding variable with you chosen value. 3. Solve the equation for the other variable.

    17. Graphing Linear Equations 1. Find at least two solutions to the equation. 2. Plot the solutions as points in the rectangular coordinate system. 3. Connect the points to form a straight line.

    18. Horizontal Line y = constant Example: y = 4 y-intercept (0,4) Function no x intercept

    19. Vertical Line x = constant Example x = -5 x-intercept (-5,0) No y intercept Not a function

    20. Objective: Know the slope formula

    21. Slope

    22. Objective: Given two points, determine the slope of a line.

    23. Slope formula

    24. Horizontal line y = constant Slope is 0 Examples: y = 5 y = -3 Can be done with calculator.

    25. Vertical Line x=constant Undefined slope Examples: x =2 x = -3 Not graphed by calculator

    26. Objective: Know and use the slope-intercept formula

    27. Slope Intercept Form for equation of Line y=mx+b Slope is m y-intercept is (0,b)

    28. Using Slope Intercept form to graph a line 1. Write the equation in form y=mx+b 2. Plot y intercept (0,b) 3. Write slope with numerator as positive or negative 3. Use slope move up or down from y intercept and then right- plot point. 4. Draw line through two points.

    29. Problem The percentage B of automobiles with airbags can be modeled by the linear function B(t)-5.6t 3.6, where t is the number of years since 1990. What is the slope of the graph of B? Answer is 5.6

    30. Objective Use slope-intercept form to write the equation of a line.

    31. y=mx+b Write the equation of a line given the slope and the y intercept. Line slope is 2 and y intercept (0,-3) y=2x-3

    32. y=mx+b Write the equation of a line given the slope and one point. Slope of 2 and point (1,3) y=2x+1

    33. Fred Couples Professional Golfer When youre prepared youre more confident: when you have a strategy youre more comfortable.

    34. Objective: Know and use the Point slope Formula

    35. Point-slope form of Linear equation

    36. Objective: Write equation of a line given the slope and one point Problem: slope of 3 through (2,-4) Answer: y=-3x+2

    37. Objective: Write equation of a line given the slope and one point Problem: slope of 3 through (2,-4) Answer: y=-3x+2

    38. Objective: Know and use the general form for the equation of a line

    39. Objectives: Determine if two lines are parallel. Determine if two lines are perpendicular.

    40. Objective: Know what determines if lines are parallel

    41. Def: Parallel Lines Two distinct non-vertical lines are parallel if and only if they have the same slope. Two distinct vertical lines are parallel.

    42. Def: Parallel Lines Two distinct non-vertical lines are parallel if and only if they have the same slope. Two distinct vertical lines are parallel.

    43. Def 1: Perpendicular Lines Two distinct lines are perpendicular if and only if the product of their slopes is 1. A vertical line and horizontal line are perpendicular.

    44. Def 2: Perpendicular Lines The slope of a line perpendicular to another line is the negative reciprocal of the slope of the original line. If slope is a/b, slope of perpendicular line is b/a.

    45. Def: Rate of Change For a linear equation in two variables, the rate of change of y with respect to x is the slope of the graph of the equation. This rate of change is a constant.

    46. Objective: Know the two intercept form for the equation of a line.

    47. Objective: Know what determines if lines are perpendicular

    48. Find the equation of a line given specific information Given two points Given a point and the slope Given a point and the equation of a line find the equation of line parallel or perpendicular to the given line.

    49. Mathematics 116 2.6 Explore data: Linear Models and Scatter Plots

    50. Objectives: Use the calculator to determine linear models for data. Graph linear model and scatter plot Make predictions based on model

    51. Objectives Construct Scatter Plots By hand With Calculator Interpret correlation Positive Negative No discernible correlation

    52. Objectives: Use the calculator to determine linear models for data. Graph linear model and scatter plot Make predictions based on model

    54. COLLEGE ALGEBRA Introduction To Linear Equations

    55. Def: Equation An equation is a statement that two algebraic expressions have the same value.

    56. Def: Solution Solution: A replacement for the variable that makes the equation true. Root of the equation Satisfies the Equation Zero of the equation

    57. Def: Solution Set A set containing all the solutions for the given equation. Could have one, two, or many elements. Could be the empty set Could be all Real numbers

    58. Def: Linear Equation in One Variable An equation that can be written in the form ax + b = c where a,b,c are real numbers and a is not equal to zero

    59. Linear function A function of form f(x) = ax + b where a and b are real numbers and a is not equal to zero.

    60. Equation Solving: The Graphing Method 1. Graph the left side of the equation. 2. Graph the right side of the equation. 3. Trace to the point of intersection Can use the calculator for intersect The x coordinate of that point is the solution of the equation.

    61. Equation solving - graphing The y coordinate is the value of both the left side and the right side of the original equation when x is replaced with the solution. Hint: An integer setting is useful Hint: x setting of [-9.4,9.4] also useful

    62. Def: Identity An equation is an identity if every permissible replacement for the variable is a solution. The graphs of left and right sides coincide. The solution set is R

    63. Def: Inconsistent equation An equation with no solution is an inconsistent equation. Also called a contradiction. The graphs of left and right sides never intersect. The solution set is the empty set.

    64. Example SOLVE ALGEBRAICALLY AND GRAPHICALLY

    65. Example SOLVE ALGEBRAICALLY AND GRAPHICALLY

    66. Example

    67. Addition Property of Equality If a = b, then a + c = b + c For all real numbers a,b, and c. Equals plus equals are equal.

    68. Multiplication Property of Equality If a = b, then ac = bc is true For all real numbers a,b, and c where c is not equal to 0. Equals times equals are equal.

    69. Solving Linear Equations Simplify both sides of the equation as needed. Distribute to Clear parentheses Clear fractions by multiplying by the LCD Clear decimals by multiplying by a power of 10 determined by the decimal number with the most places Combine like terms

    70. Solving Linear Equations Cont: Use the addition property so that all variable terms are on one side of the equation and all constants are on the other side. Combine like terms. Use the multiplication property to isolate the variable Verify the solution

    71. FORMULAS Solve Formulas Isolate a particular variable in a formula Treat all other variables like constants Isolate the desired variable using the outline for solving equations.

    72. Section 3.3 Solve Formulas Isolate a particular variable in a formula Treat all other variables like constants Isolate the desired variable using the outline for solving equations.

    73. Formulas continued Area of a square Perimeter of a square

    74. Formulas continued Area of Parallelogram A = bh

    75. Formulas continued Trapezoid

    76. Formulas continued Area of Circle Circumference of Circle

    77. Formulas continued: Area of Triangle

    78. Formulas continued Sum of measures of a triangle

    79. Formulas continued Perimeter of a Triangle

    80. Formulas continued Pythagorean Theorem

    81. Formulas continued: Volume of a Cube all sides are equal

    82. Formulas continued Rectangular solid Area of Base x height

    83. Formulas continued Volume Right Circular Cylinder

    84. Formulas continued: Surface are of right circular cylinder

    85. Formulas continued: Volume of Right Circular Cone V=(1/3) area base x height

    86. Formulas continued: Volume Sphere

    87. Formulas continued: General Formula surface area right solid SA = 2(area base) + Lateral surface area SA=2(area base) + LSA Lateral Surface Area = LSA = (perimeter)*(height)

    88. Formulas continued: General Formula surface area right solid SA = 2(area base) + Lateral surface area SA=2(area base) + LSA Lateral Surface Area = LSA = (perimeter)*(height)

    89. Useful Calculator Programs CIRCLE CIRCUM CONE CYLINDER PRISM PYRAMID TRAPEZOI APPS-AreaForm

    90. Robert Lewis Stevenson Dont judge each day by the harvest you reap, but by the seeds you plant.

    91. Solve by Graphing Graph the left and right sides and find the point of intersection Determine where x values are above and below. Solution is x values y is not critical

    92. Example solve by graphing

    93. Addition Property of Inequality If a < b, then a + c = b + c for all real numbers a, b, and c

    94. Multiplication Property of Inequality For all real numbers a,b, and c If a < b and c > 0, then ac < bc If a < b and c < 0, then ac > bc

    95. Compound Inequalities 3.7 Def: Compound Inequality: Two inequalities joined by and or or

    96. Intersection - Disjunction Intersection: For two sets A and B, the intersection of A and B, is a set containing only elements that are in both A and B.

    97. Solving inequalities involving and 1. Solve each inequality in the compound inequality 2. The solution set will be the intersection of the individual solution sets.

    98. Solving inequalities involving and 1. Solve each inequality in the compound inequality 2. The solution set will be the intersection of the individual solution sets.

    99. Solving inequalities involving or Solve each inequality in the compound inequality The solution set will be the union of the individual solution sets.

    100. Knute Rockne Notre Dame football coach 1888-1931 Build up your weaknesses until they become your strengths.

    101. Walter Elliott Perseverance is not a long race. It is many short races one after another.

    102. Abraham Lincoln U.S. President Nothing valuable can be lost by taking time.

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